LONDON (Reuters) - A post-Brexit trade deal with the United States would not be enough to make up for leaving the European Union, British justice minister David Lidington said on Sunday, tempering Prime Minister Theresa May's enthusiasm about the U.S. offer.

May had warmly welcomed assurances on Saturday by U.S. President Donald Trump that a "very powerful" trade deal with Britain would be reached "very, very quickly" after Britain leaves the EU.

Seeking to reassert her authority over a Brexit process thrown into chaos by a botched snap election last month, May described talks on trade with Trump and other world leaders at a G20 meeting as a "powerful vote of confidence" in Britain.

But one of her senior ministers dampened that enthusiasm on Sunday, in a sign of the difficult task May faces in uniting her own party behind a single exit strategy as key legislation is due to enter parliament next week.

"It wouldn't be enough on its own, no," Lidington told the BBC's Andrew Marr Show. "But it would be a very good thing to have - as would trade deals with the emerging economies of Asia and Latin America."

The government has touted the ability to strike bilateral trade deals, rather than EU-wide deal negotiated by Brussels, as one of the key benefits of leaving the bloc.

Lidington campaigned strongly for Britain to remain in the EU before the 2016 referendum, but has since said he accepts the outcome of that vote.

The Confederation of British Industry, a business lobby group, also warned against placing too much emphasis on a potential deal with the United States.

"Not every trade deal is necessarily a good and fair trade deal for both parties," CBI President Paul Drechsler told Sky television.

"The USA has one of the best negotiating teams in the world in terms of trade deals. We don't want to walk into a bear hug - I would be wary of trying to be too fast on a trade deal."

LEADERSHIP RELAUNCH

May is expected to make a speech on Tuesday marking a year since she inherited power following Britain's surprise vote to leave the EU.

The speech is seen as an attempt to move on from a tumultuous 12 months in which May set out a hardline approach to leaving the EU and called an election, but then failed to persuade the public to back her and lost her outright majority in parliament.

She has struck a deal with a small Northern Irish political party to support her minority government.

Her weakened status has led to speculation that members of her Conservative Party - historically split between eurosceptics and more pro-European members - would be prepared to oust her if they did not get their way.

Party unity will be tested by legislation which is due to be presented to parliament next week, outlining the government's plan to translate all EU law in British law as part of its preparation for leaving.

A report in the Sunday Times newspaper cited anonymous sources as saying pro-European lawmakers could demand concessions in the bill.

A separate report sourced to an unnamed Conservative in the Mail on Sunday also said a 'Kamikaze' group of Brexit-supporting Conservatives would be prepared to oust her and force a new election if she did water down her Brexit plans.

Dismissing those reports, Lidington said: "I’ve been in parliament 25 years and almost every July a combination of too much sun and too much warm prosecco leads to gossipy stories in the media."